Silas Johnson on weighted discriminants with mass formulas

How should you count number fields? The most natural way is by discriminant; you count all degree-n number fields K with a given Galois group G in S_n and discriminant bounded in absolute value by B. This gives you a value N_G(B) whose asymptotic behavior in B you might want to study. The classical results and exciting new ones you’ve heard about — Davenport-Heilbron, Bhargava, and all that — generally concern counts of this kind.

But there are reasons to consider other kinds of counts. I once had a bunch of undergrads investigate the behavior of N_3(X,Y), the number of cubic fields whose discriminant had squarefree part at most X and maximal square divisor at most Y. This gives a more refined picture of the ramification behavior of the fields. Asymptotics for this are still unknown! (I would expect the main term to be on order , but I don’t know what the secondary terms should be.)

One nice thing about the discriminant, though, is that it has a mass formula. In brief: a map f from Gal(Q_p) to S_n corresponds to a degree-n extension of Q_p, which has a discriminant (a power of p) which we call Disc(f). Averaging Disc(f)^{-1} over all homomorphisms f gives you a polynomial in p^{-1}, which we call the local mass. Now here’s the remarkable fact (shown by Bhargava, deriving from a formula of Serre) — there is a universal polynomial P(x) such that the local mass at p is equal to P(p^{-1}) for every P. This is not hard to show for the tame primes p (you can see this point discussed in Silas’s paper or in the paper by Kedlaya I linked above) but that it holds for the wild primes is rather mysterious and strange. And it certainly seems to ratify the idea that there’s something especially nice about the discriminant. What’s more, this polynomial P is not just some random thing; it’s the product over p of P(p^{-1}) that gives the constant in Bhargava’s conjectural asymptotic for the number of number fields for degree n.

But here’s the thing. If we replace G by a subgroup of S_n, there need not be a universal mass formula anymore. Kedlaya (and Daniel Gulotta, in the appendix) show lots of examples. The simplest example is the dihedral group of order 8.

All is not lost, though! Wood showed in 2008 that you could fix this problem by replacing the discriminant of a D_4-extension with a different invariant. Namely: a D_4 quartic field M has a quadratic subextension L. If you replace Disc(L/Q) with Disc(L/Q) times the norm to Q of Disc(L/M), you get a different invariant of M — an example of what Silas calls a “weighted discriminant” — and when you compute the local mass according to {\em this} invariant, you get a polynomial in p^{-1} again!

So maybe Wood’s modified discriminant, not the usual discriminant, is the “right” way to count dihedral quartics? Does the product of her local masses give the right asymptotic for the number of D_4 extensions with Woodscriminant at most B?

It’s not at all clear to me how, if at all, you can cook up a modified discriminant for an arbitrary group G that has a universal mass formula. What Silas shows is that having a mass formula is indeed special; when G is a p-group, there are only finitely many weighted discriminants that have one. Silas thinks, and so do I, that this is actually true for every finite group G, and that some version of his approach will eventually prove this. And he classifies all such weighted discriminants for groups of size up to 12; for any individual G, it’s a computation which can be made nicely algorithmic. Very cool!